Ellipsoidal Approximation Research Articles

DEVELOPMENT OF THE GUARANTEED ESTIMATION ROBUST ALGORITHM OF LINEAR CONTROLLED SYSTEM STATES

Ellipsoidal approximation of the ellipsoid and hyperlayer crossing has been considered as a basis of the algorithm of states estimation of the linear controlled system whose set of possible states is represented with an ellipsoid, and observations – with a hyperlayer. This representation is considered as an analogue of Kalman filter. The conditions of a priori system state and a posteriori measurement information compatibility and sensitivity of the algorithm to a choice of its parameters have been investigated. Dependence of the system state estimate improvement on a relative width of the hyperlayer of a set of observations has been shown. The obtained algorithm in comparison with the known solutions at minor degradation of accuracy is much easier in realization and stabler in operation from the standpoint of prior guesses violation.

The representation of asteroid shapes: A test for the inversion of Gaia photometry

It is common practice nowadays to derive spins and 3D shapes of asteroids from the inversion of photometric light curves. However, this method requires, in general, a good number of photometric points and dedicated observing sessions. On the other hand, the photometric observations carried out by the Gaia mission will be sparse and their number relatively small.For this reason, a multi-parametric shape described by a large number of elementary facets cannot probably be derived from Gaia data alone. Therefore, the Data Processing and Analysis Consortium (DPAC), implemented a simpler solution as an unattended data reduction pipeline, which relies on three axial ellipsoids for the shape representation.However, overall accuracy of such triaxial shape solutions has to be assessed. How adequate is an ellipsoidal approximation to represent the overall properties of an irregular body? Which error is made on the volume in comparison to a more accurate model with irregular shape?To answer these questions, we have implemented numerical procedures for comparing ellipsoids to more complex and irregular shapes, and we performed a full simulation of the photometric signal from these objects, using both shape representations. Implementing the same software algorithm that will be used for the analysis of Gaia asteroid photometry, rotation period, spin axis orientation and ellipsoidal shape were derived from simulated observations of selected Main Belt asteroids assuming a geometric scattering model (work is in progress for more complex scattering models).Finally, these simulated Gaia results were compared to check the relevance of the ellipsoidal solution in comparison to multi-parametric shapes. We found that the ellipsoids by photometry inversion are closely similar to the best-fitting ellipsoids of the simulated complex shapes and that the error on the asteroid volume (relative to a complex shape) is generally low, usually around 10%.

로버스트주성분회귀에서 최적의 주성분선정을 위한 기준

회귀모형에 연관성이 높은 설명변수들이 포함되면 다중공선성의 문제가 야기되며, 동시에 자료에 회귀 이상점들이 포함되면 최소자승추정량에 바탕을 둔 제반 통계적 추론은 심각한 결함을 갖게 된다. 이러한 현상들은 데이터마이닝 분야에서 많이 볼 수 있는데, 본 논문에서는 두 가지 문제를 동시에 해결하기 위한 방안으로서 로버스트주성분회귀를 제안하였다. 특히 최적의 주성분을 선정하기 위한 새로운 기준을 개발하였는데, 설명변수들의 표본공분산 대신에 MVE-추정량을 기반으로 하였으며, 고유치가 아니라 상태지수의 크기에 바탕을 둔 선정기준을 제안하였다. 그리고 주성분모형에서의 추정을 위하여 회귀이상점에 대해 로버스트한 LTS-추정을 도입하였다. 제안된 선정기준이 기존의 기준들보다 다중공선성과 이상점이 유발하는 문제들을 잘 해결할 수 있음을 모의실험을 통하여 확인하였다. Robust principal components regression is suggested to deal with both the multicollinearity and outlier problem. A main aspect of the robust principal components regression is the selection of an optimal set of principal components. Instead of the eigenvalue of the sample covariance matrix, a selection criterion is developed based on the condition index of the minimum volume ellipsoid estimator which is highly robust against leverage points. In addition, the least trimmed squares estimation is employed to cope with regression outliers. Monte Carlo simulation results indicate that the proposed criterion is superior to existing ones.

Measuring volume and density of a barley grain using ellipsoid approximation from a 2-D digital image

A digital imaging method was developed to determine the size, weight, volume and density of individual barley grains. Calibrations were developed using morphological measurements based on 2-D digital images, and were validated using a range of bulk barley samples. The validation of the grain diameter (minor) and grain weight algorithms gave high correlation coefficients ( r = 0.93 and 0.97, respectively, P < 0.01), with low standard errors (0.09 mm and 1.37 mg, respectively) when compared to physical measurements. Estimated grain volume and density from image data, using three different mathematical approaches were compared: Model 1, a general ellipsoid; Model 2, a combination of two cone frustums with an ellipse top and base; and Model 3, a set of elliptical cylinders representing pixels along the image length. Derived grain volume data using Model 2 when compared to gas displacement volume measurements gave significant correlations ( r = 0.97, P < 0.01) with the lowest standard error (0.97 mm 3), in contrast to volume data estimated using Model 1 and Model 3. Even though Model 2 gave accurate and precise results for grain volume, Model 1 (with the slope adjusted) gave the highest correlation ( r = 0.63, P < 0.01) with the lowest standard error (0.055 mg/mm 3) for estimating grain density from image data, compared to gas displacement density measurements.

Minimum-volume ellipsoidal approximation of the sum of two ellipsoids

The problem of the ellipsoidal approximation of the sum of two ellipsoids optimal with respect to the minimum of multidimensional volume is considered. It is solved without the use of affinities and representation as a conditional optimization problem. The case of simultaneous degeneracy of the ellipsoids is considered. A geometrical interpretation of the approximation is given. Results of the numerical modeling are presented.

Constructing an analogue of the Kalman filter for a guaranteed state estimation of a dynamic system

A filter similar to the known Kalman filter is constructed to obtain the external guaranteed ellipsoidal estimate of the state of a dynamic system using measurements. Information on uncertain factors acting upon the system and observation errors is supposed to be restricted to knowing the boundaries of possible values of these variables. A numeric example of how the obtained relations can be used is given.

Algorithms to estimate the reachability sets of the pulse controlled systems with ellipsoidal phase constraints

Methods to construct the ellipsoidal estimates of the reachability sets of a nonlinear dynamic system with scalar pulse control and uncertainty in the initial data were proposed. The considered pulse system was rearranged in the ordinary differential inclusion already without any pulse components by means of a special discontinuous time substitution. The results of the theory of ellipsoidal estimation and the theory of evolutionary equations of the multivalued states of dynamic systems under uncertainty and ellipsoidal phase constraints were used to estimate the reachability sets of the resulting nonlinear differential inclusion.

Ellipsoidal area mean gravity anomalies — precise computation of gravity anomaly reference fields for remove-compute-restore geoid determination

Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available as area mean values.

Ground deformation in the Taupo Volcanic Zone, New Zealand, observed by ALOS PALSAR interferometry

SUMMARY We present ground deformation measurements in the Taupo Volcanic Zone (TVZ) using differential interferomeric synthetic aperture radar (DInSAR) observations collected by ALOS PALSAR during 2006–2010, and compare them with displacement observations from continuous GPS. We acquired and processed DInSAR images from two ascending paths (324 and 325) and one descending path (628) covering the TVZ, and produced linear deformation rates and time series of deformation. The DInSAR results were improved by using a modified version of the small baseline subset (SBAS) algorithm that simultaneously solves for deformation rates and residual topographic noise. The accuracy of the DInSAR displacement rates along line-of-sight to the satellite is 0.5–2 cmyr −1 depending on the number of SAR images and their coherence. We found good agreement between the DInSAR-derived displacement rates and those measured by continuous GPS for the two ascending paths (correlation 0.94 ± 0.01 and 0.89 ± 0.02); the DInSAR uncertainties were too large to make a useful comparison for the descending path (correlation 0.66 ± 0.03). We identified ground deformation due to groundwater and steam extraction for geothermal power. To demonstrate the geophysical application, we modelled the deformation results using simplified sources for some of the geothermal signals using ellipsoidal and tabular approximations.

Development of casting shrinkage cavity modelling toolkit for finite element analysis using ellipsoidal approximation technique

This paper presents a technique for automatically approximating casting shrinkage cavity models and its application in finite element analysis. First, reverse engineering was utilised to obtain the stereolithography format models of internal shrinkage cavities. After that, an in-house python code was used to analyse the shape of shrinkage cavity and then decompose it to ellipsoids, whose volume sum covers all points of the given shrinkage cavity obtained with reverse engineering. Finally, the ellipsoids were merged into a single ellipsoidal blob shaped three-dimensional computer aided design model, which can be used to model the casting with internal shrinkage for finite element analysis. The code was developed and implemented into ABAQUS as a customised plug-in toolkit, which makes it easy for engineers to model the casting with internal shrinkage cavities.

Use of Green’s function for determining the disturbing potential of an ellipsoidal Earth

A spherical approximation makes the basis for a majority of formulas in physical geodesy. However, the present-day accuracy in determining the disturbing potential requires an ellipsoidal approximation. The paper deals with constructing Green’s function for an ellipsoidal Earth by an ellipsoidal harmonic expansion and using it for determining the disturbing potential. From the result obtained the part that corresponds to the spherical approximation has been extracted. Green’s function is known to depend just on the geometry of the surface where boundary values are given. Thus, it can be calculated irrespective of the gravity data completeness. No changes of gravity data have an effect on Green’s function and they can be easily taken into account if the function has already been constructed. Such a method, therefore, can be useful in determining the disturbing potential of an ellipsoidal Earth.

Toward accurate volumetry of brain aneurysms: Combination of an algorithm for automatic thresholding with a 3D eraser tool

The present study describes a new approach for aneurysm volume quantification on three-dimensional angiograms, which focuses on solving three common technical problems: the variability associated with the use of manual thresholds, the irregular morphology of some aneurysms, and the imprecision of the limits between the parent artery and the aneurysm sac. The method consists of combining an algorithm for automatic threshold determination with a spherical eraser tool that allows the user to separate the image of the aneurysm from the parent artery. The accuracy of volumetry after automatic thresholding was verified with an in vitro experiment in which 57 measurements were performed using four artificial aneurysms of known volume. The reliability of the method was compared to that obtained with the technique of ellipsoid approximation in a clinical setting of 15 real angiograms and 150 measurements performed by five different users. The mean error in the measurement of the artificial aneurysms was 7.23%. The reliability of the new approach was significantly higher than that of the ellipsoid approximation. Limits of agreement between two measurements were determined with Bland–Altman plots and ranged from −14 to 13% for complex and from −10.8 to 11.03% for simple-shaped sacs. The reproducibility was lower (>20% of variation) for small aneurysms (<70 mm 3) and for those presenting a very wide neck (dome-to-neck ratio < 1). The method is potentially useful in the clinical practice, since it provides relatively precise, reproducible, volume quantification. A safety coiling volume can be established in order to perform sufficient but not excessive filling of the aneurysm pouch.

2214 AN ELLIPSOID MODEL OF THE KIDNEY CALCULATED FROM MULTIDETECTOR COMPUTED TOMOGRAPHY PREDICTS DIFFERENTIAL FUNCTION IN LIVING RELATED DONORS

You have accessJournal of UrologyImaging/Radiology: Uroradiology1 Apr 20112214 AN ELLIPSOID MODEL OF THE KIDNEY CALCULATED FROM MULTIDETECTOR COMPUTED TOMOGRAPHY PREDICTS DIFFERENTIAL FUNCTION IN LIVING RELATED DONORS John Henderson, Maitham Moslim, Matthew Knox, and Nigel Cowan John HendersonJohn Henderson High Wycombe, United Kingdom More articles by this author , Maitham MoslimMaitham Moslim Damascus, Syrian Arab Republic More articles by this author , Matthew KnoxMatthew Knox Calgary, Canada More articles by this author , and Nigel CowanNigel Cowan Oxford, United Kingdom More articles by this author View All Author Informationhttps://doi.org/10.1016/j.juro.2011.02.2454AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookTwitterLinked InEmail INTRODUCTION AND OBJECTIVES Computed tomography and nuclear medicine studies are routinely used for the assessment of renal anatomy and function respectively. Assessment of renal volume and attenuation on CT for diseased kidneys has been shown to allow accurate calculation of differential renal function. We considered whether a simple ellipsoid approximation of renal volume, calculated from 3-axis renal measurements would predict differential renal function in healthy kidneys. METHODS Potential living related donors (LRD) were evaluated with triple-bolus multidetector computed tomography (TB-MDCT) and technetium-99m dimercaptosuccinic acid renography (Tc-99m DMSA). Images were independently assessed by two observers, both blinded to patient demographics and result of DMSA renography. Three dimensional reconstruction was undertaken and images analysed in the sagittal and axial planes. The maximal renal diameter for the long-axis of the kidney in the sagittal plane was obtained (d1). A perpendicular line was drawn from this at the point of maximum anteroposterior depth (d2). The maximum diameter in the coronal plane (d3) was measured on the axial images. An ellipsoid model of the kidney was employed, using the formula; Renal volume= 4/3 x Π x ½d1 x ½d2 x ½d3 The volume of each left and right kidney were summated to give total renal volume. Each renal volume was divided by total renal volume x 100 to give percentage renal volume. The degree of interobserver variability and correlation of ellipsoid volume modelling with DMSA derived DRF was tested with the Pearson correlation coefficient and statistical significance taken as p<0.05. RESULTS 50 patients underwent TB-MDCT and DMSA. All examinations were technically adequate. The correlation of TB-MDCT and DMSA was good with p<0.001 for both left and right renal volumes. Renal volume as measured by two observers correlated closely for both left and right renal volumes (P<0.0001). CONCLUSIONS An ellipsoid approximation of renal volume calculated from renal measurements correlates closely with differential renal function obtained from radionuclide studies. This method is simple and reproducible between observers. Obtaining both anatomical and functional information from a single study saves time and resources whilst exposing the patient to less radiation. © 2011 by American Urological Association Education and Research, Inc.FiguresReferencesRelatedDetails Volume 185Issue 4SApril 2011Page: e888-e889 Advertisement Copyright & Permissions© 2011 by American Urological Association Education and Research, Inc.MetricsAuthor Information John Henderson High Wycombe, United Kingdom More articles by this author Maitham Moslim Damascus, Syrian Arab Republic More articles by this author Matthew Knox Calgary, Canada More articles by this author Nigel Cowan Oxford, United Kingdom More articles by this author Expand All Advertisement Advertisement PDF downloadLoading ...

New robust estimators for detecting non-random patterns in multivariate control charts: a simulation approach

In the past decade, different robust estimators have been proposed by several researchers to improve the ability to detect non-random patterns such as trend, process mean shift, and outliers in multivariate control charts. However, the use of the sample mean vector and the mean square successive difference matrix in the T 2 control chart is sensitive in detecting process mean shift or trend but less sensitive in detecting outliers. On the other hand, the minimum volume ellipsoid (MVE) estimators in the T 2 control chart are sensitive in detecting multiple outliers but less sensitive in detecting trend or process mean shift. Therefore, new robust estimators using both merits of the mean square successive difference matrix and the MVE estimators are developed to modify Hotelling's T 2 control chart. To compare the detection performance among various control charts, a simulation approach for establishing control limits and calculating signal probabilities is provided as well. Our simulation results show that a multivariate control chart using the new robust estimators can achieve a well-balanced sensitivity in detecting the above-mentioned non-random patterns. Finally, three numerical examples further demonstrate the usefulness of our new robust estimators.

Construction of the best ellipsoidal approximation of the attraction domain in stabilization problem for a wheeled robot

The synthesis control problem for the plane motion of a wheeled robot with constrained control resource is studied. The goal of the control is to bring the robot to an assigned curvilinear trajectory and to stabilize its motion along it. For a synthesized control law, the problem of finding the best in the sense of volume ellipsoidal approximation of the attraction domain of the target path is posed. To take into account constraints on the control, an approach based on methods of absolute stability theory is used, in the framework of which construction of an approximating ellipsoid reduces to solving a system of linear matrix inequalities. It is shown that the desired maximum-volume approximating ellipsoid can be found by solving a standard constrained optimization problem for a function of two variables.

摩擦円すいの楕円体近似を用いた把持安定性の高速評価

This paper presents a fast method of evaluating grasp stability when a multi-fingered hand grasps objects. Different from previous methods, we consider approximating the friction cone by using a few ellipsoids. By using this method, the total force/moment set applied to the grasped object can be obtained by a common set of multiple ellipsoids. This method is not conservative but effective since approximated force closure is tested by simply calculating the inequalities of quadratic form. Moreover, by using the ellipsoidal approximation, we propose an easy method of evaluating the grasp stability. We show that the grasp stability can be easily calculated by using simple equations. The effectiveness of the proposed method is verified by several numerical examples where we show that the proposed method has practical accuracy for real applications and evaluates the grasp stability much faster than conventional methods.

Stability Analysis of Nonlinear Switched Systems via Reduction Method

AbstractStability of nonlinear switched systems is discussed. A method of constructing the common (scalar and vector) Lyapunov functions is proposed for switched systems whose equations have homogenous right-hand sides and for switched large-scale systems containing homogeneous subsystems. With the aid of homomorphic mapping the problem of stability for nonhomogeneous nonlinear systems is reduced to an auxiliary problem, which is easier to analyze. Matrix homomorphism is used for the purpose of ellipsoidal estimation of nonlinear dynamics of the polynomial class of systems. Certain applications are mentioned.

Planar, spherical and ellipsoidal approximations of Poisson's integral in near zone

Planar, spherical and ellipsoidal approximations of Poisson's integral in near zonePlanar, spherical, and ellipsoidal approximations of Poisson's integral for downward continuation (DWC) of gravity anomalies are discussed in this study. The planar approximation of Poisson integral is assessed versus the spherical and ellipsoidal approximations by examining the outcomes of DWC and finally the geoidal heights. We present the analytical solution of Poisson's kernel in the point-mean discretization model that speed up computation time 500 times faster than spherical Poisson kernel while preserving a good numerical accuracy. The new formulas are very simple and stable even for regions with very low height. It is shown that the maximum differences between spherical and planar DWC as well as planar and ellipsoidal DWC are about 6 mm and 18 mm respectively in the geoidal heights for a rough mountainous area such as Iran.

On the use of minimum volume ellipsoids and symplectic capacities for studying classicaluncertainties for joint position–momentum measurements

We study the minimum volume ellipsoid estimator associated with a cloud of points inphase space. Using as a natural measure of uncertainty the symplectic capacity of thecovariance ellipsoid we find that classical uncertainties obey relations similar to those foundin non-standard quantum mechanics.

Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system

The problem of estimating trajectory tubes of a nonlinear control dynamical system with uncertainty in initial data is considered. It is assumed that the dynamical system has a special structure, in which nonlinear terms are quadratic in phase coordinates and the values of the uncertain initial states and admissible controls are subject to ellipsoidal constraints. Differential equations are found that describe the dynamics of the ellipsoidal estimates of reachable sets of the nonlinear dynamical system under consideration. To estimate reachable sets of the nonlinear differential inclusion corresponding to the control system, we use results from the theory of ellipsoidal estimation and the theory of evolution equations for set-valued states of dynamical systems under uncertainty.